The tame Breuil–Mézard conjecture for unramified groups

Let $G$ be an unramified group over $\mathbf{Q}_p$ satisfying Hypothesis, and let $d=\dim\check{\mathcal{B}}$ be the dimension of the flag variety of the dual group. For each Serre weight $\sigma$ of $G(\mathbf{F}_p)$, let $n_\sigma(\lambda,\tau)=[\overline{W(\lambda)\otimes\sigma(\tau)}:\sigma]$. Tame Breuil–Mézard conjecture. There is a collection of effective cycles $\mathcal{Z}(\sigma)\in Z_d((\mathcal{X}_{{}^{L}\!G}^{\operatorname{EG}})_{\operatorname{red}})$ such that, for every dominant weight $\lambda$ and every tame inertial parameter $\tau$,

$$[\mathcal{X}^{\lambda+\rho,\tau}_{{}^{L}\!G,\mathbf{F}}]=\sum_\sigma n_\sigma(\lambda,\tau)\mathcal{Z}(\sigma)\in Z_d((\mathcal{X}_{{}^{L}\!G}^{\operatorname{EG}})_{\operatorname{red}}).$$

This is the tame part of the expected Breuil–Mézard conjecture for general unramified groups. The full formulation is currently obstructed by the lack of a sufficiently general inertial local Langlands correspondence.